A clothing manufacturer has revenue and cost functions, both in dollars, given R(x)=35x-0.1x^2 and C(x)=4x+2000, respectively , where x is the number of t-shirts sold. A) how many t-shirts must the company sell to maximize its profit? B) what is the company’s maximum profit? 

To solve this problem, we first need to define the profit function. The profit $ P(x) $ is given by the revenue function $ R(x) $ minus the cost function $ C(x) $:

\[
P(x) = R(x) - C(x)
\]

Substituting the given functions:

\[
P(x) = (35x - 0.1x^2) - (4x + 2000)
\]

Simplifying this:

\[
P(x) = 35x - 0.1x^2 - 4x - 2000
\]

\[
P(x) = 31x - 0.1x^2 - 2000
\]

Next, we find the value of $ x $ that maximizes the profit by taking the derivative of $ P(x) $ and setting it to zero:

\[
P'(x) = 31 - 0.2x
\]

Setting the derivative equal to zero:

\[
31 - 0.2x = 0
\]

Solving for $ x $:

\[
0.2x = 31 \implies x = \frac{31}{0.2} = 155
\]

So, **A)** the company must sell **155 t-shirts** to maximize its profit.

Next, to find the maximum profit, we substitute $ x = 155 $ back into the profit function:

\[
P(155) = 31(155) - 0.1(155^2) - 2000
\]

Calculating $ P(155) $:

1. Calculate $ 31 \times 155 = 4805 $
2. Calculate $ 155^2 = 24025 $ and then $ 0.1 \times 24025 = 2402.5 $
3. Now substitute into the profit function:

\[
P(155) = 4805 - 2402.5 - 2000
\]

\[
P(155) = 4805 - 4402.5 = 402.5
\]

So, **B)** the company’s maximum profit is **$402.50**.

### Summary:
- **A)** Number of t-shirts to maximize profit: **155**
- **B)** Maximum profit: **$402.50**

Would you like more details on any part of this problem, or do you have any questions? Here are some related questions you might consider:

1. How would the profit change if the cost function increased?
2. What is the break-even point for the company?
3. How do changes in the price of t-shirts affect revenue?
4. What is the impact of selling fewer than 155 t-shirts on profit?
5. How can the company lower costs to increase profit?

**Tip:** To maximize profit, always analyze the relationship between revenue and cost functions, as small changes can significantly impact overall profitability.

A clothing manufacturer has revenue and cost functions, both in dollars, given R(x)=35x-0.1x^2 and C(x)=4x+2000, respectively, where x is the number of t-shirts sold. A) how many t-shirts must the company sell to maximize its profit? B) what is the company’s maximum profit?

Learn how to calculate the maximum profit for a clothing manufacturer with revenue function R(x) = 35x - 0.1x^2 and cost function C(x) = 4x + 2000. Understand profit maximization using calculus, derivatives, and algebraic simplification. Ideal for Grades 11-12.

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